Group #22590393 — subgroups

Subgroups — #22590393

Order 9088.

170 conjugacy classes of subgroups; only the 49 normal subgroups are shown.

normal subgroups only

This group has 170 conjugacy classes of subgroups — too many to draw. Only its 49 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#14556072	C₂	1	1	yes	maximal, characteristic, complemented
2	#17986382	C₂	1	1	yes	maximal, characteristic, complemented
3	D₈×Dic₇₁	C₂	1	1	yes	maximal, characteristic, complemented
4	Dic₇₁⋊C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
5	#17985804	C₂	1	1	yes	maximal, characteristic, complemented
6	#17985800	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×D₁₆×C₇₁	C₂	1	1	yes	maximal, characteristic
8	C₂×Dic₂₈₄	C₂²	1	1	yes	characteristic
9	C₂×#11765853	—	2	1	no	—
10	#14556115	—	2	1	no	—
11	D₄×Dic₇₁	—	2	1	no	—
12	#14556072	—	2	2	no	—
13	C₂×C₇₁⋊C₁₆	C₂²	1	1	yes	characteristic
14	#14556062	C₂²	1	1	yes	characteristic
15	#14556049	—	2	1	no	—
16	C₈×Dic₇₁	C₂²	1	1	yes	characteristic
17	C₂×D₈×C₇₁	C₂²	1	1	yes	characteristic
18	C₂×D₈×C₇₁	C₂²	1	1	yes	characteristic
19	D₁₆×C₇₁	—	2	2	no	—
20	C₂×C₁₆×C₇₁	C₂²	1	1	yes	characteristic
21	C₂×Dic₁₄₂	—	2	1	no	—
22	C₂²×Dic₇₁	—	4	1	no	—
23	Dic₂₈₄	—	2	1	no	—
24	Dic₂₈₄	—	2	1	no	—
25	#11765853	—	4	2	no	—
26	C₂×C₇₁⋊C₈	D₄	1	1	yes	characteristic
27	C₄⋊Dic₇₁	—	2	1	no	—
28	C₂²⋊Dic₇₁	—	4	1	no	—
29	C₂²⋊Dic₇₁	—	4	1	no	—
30	C₂²⋊Dic₇₁	—	4	1	no	—
31	C₄×Dic₇₁	D₄	1	1	yes	characteristic
32	C₇₁⋊C₁₆	—	2	1	no	—
33	C₂×D₄×C₇₁	—	2	1	no	—
34	C₂×D₄×C₇₁	—	2	1	no	—
35	D₈×C₇₁	—	2	1	no	—
36	D₈×C₇₁	—	2	1	no	—
37	D₈×C₇₁	—	2	1	no	—
38	D₈×C₇₁	D₄	1	2	yes	—
39	C₂×C₈×C₇₁	C₂³	1	1	yes	characteristic
40	C₁₆×C₇₁	—	2	1	no	—
41	C₂³×C₇₁	—	4	1	no	—
42	C₂³×C₇₁	—	4	1	no	—
43	D₄×C₇₁	—	2	2	no	—
44	D₄×C₇₁	—	4	1	no	—
45	D₄×C₇₁	—	4	1	no	—
46	D₄×C₇₁	—	4	1	no	—
47	C₂×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
48	C₂×Dic₇₁	—	4	1	no	—
49	C₂×Dic₇₁	—	4	1	no	—
50	C₂×Dic₇₁	—	8	2	no	—
51	C₂×Dic₇₁	D₈	1	1	yes	characteristic
52	C₂×Dic₇₁	D₈	1	1	yes	characteristic
53	Dic₁₄₂	—	4	1	no	—
54	Dic₁₄₂	—	4	1	no	—
55	C₈×C₇₁	C₂×D₄	1	1	yes	characteristic
56	C₈×C₇₁	#41	1	1	yes	characteristic
57	C₇₁⋊C₈	—	2	1	no	—
58	C₂²×C₇₁	—	4	2	no	—
59	C₂²×C₇₁	—	8	1	no	—
60	C₂²×C₇₁	—	8	1	no	—
61	C₂²×C₇₁	—	8	1	no	—
62	C₂²×C₇₁	—	8	1	no	—
63	C₂²×C₇₁	—	8	1	no	—
64	C₂²×C₇₁	C₂×D₈	1	1	yes	characteristic
65	C₄×C₇₁	#121	1	1	yes	characteristic
66	C₄×C₇₁	AGL(1,ℤ/8)	1	1	yes	characteristic
67	Dic₇₁	—	2	1	no	—
68	Dic₇₁	—	2	1	no	—
69	Dic₇₁	—	8	1	no	—
70	Dic₇₁	—	8	1	no	—
71	C₂×C₇₁	—	8	2	no	—
72	C₂×C₇₁	—	16	1	no	—
73	C₂×C₇₁	#459	1	1	yes	characteristic
74	C₂×C₇₁	#508	1	1	yes	characteristic
75	C₂×C₇₁	#509	1	1	yes	characteristic
76	#2210	—	71	1	no	maximal, Sylow
77	C₇₁	#2210	1	1	yes	characteristic, complemented, Sylow
78	C₂×SD₃₂	—	71	1	no	—
79	C₂×D₁₆	D₇₁	1	1	yes	characteristic
80	#495	—	71	1	no	—
81	C₄×D₈	—	71	1	no	—
82	C₄⋊C₁₆	—	71	1	no	—
83	#358	—	71	1	no	—
84	#357	—	71	1	no	—
85	C₂×Q₁₆	—	71	1	no	—
86	C₂×SD₁₆	—	142	1	no	—
87	C₂×D₈	C₂×D₇₁	1	1	yes	characteristic
88	C₂×D₈	C₂×D₇₁	1	1	yes	characteristic
89	#124	—	142	1	no	—
90	C₄×D₄	—	142	1	no	—
91	SD₃₂	—	142	2	no	—
92	D₁₆	—	2	2	no	—
93	C₂×C₁₆	—	71	1	no	—
94	C₂×C₁₆	C₂×D₇₁	1	1	yes	characteristic
95	#107	—	71	1	no	—
96	#102	—	142	1	no	—
97	C₄×C₈	—	71	1	no	—
98	C₂×Q₈	—	142	1	no	—
99	C₂×D₄	—	2	1	no	—
100	C₂×D₄	—	2	1	no	—
101	C₂²×C₄	—	284	1	no	—
102	Q₁₆	—	142	1	no	—
103	Q₁₆	—	142	1	no	—
104	SD₁₆	—	284	2	no	—
105	D₈	—	2	1	no	—
106	D₈	—	2	1	no	—
107	D₈	—	2	1	no	—
108	D₈	C₂²⋊D₇₁	1	2	yes	—
109	C₂×C₈	—	71	1	no	—
110	C₂×C₈	C₂²×D₇₁	1	1	yes	characteristic
111	C₄⋊C₄	—	142	1	no	—
112	C₂²⋊C₄	—	284	1	no	—
113	C₂²⋊C₄	—	284	1	no	—
114	C₂²⋊C₄	—	284	1	no	—
115	C₄²	—	71	1	no	—
116	C₁₆	—	2	1	no	—
117	C₁₆	—	142	1	no	—
118	C₂³	—	4	1	no	—
119	C₂³	—	4	1	no	—
120	Q₈	—	284	1	no	—
121	Q₈	—	284	1	no	—
122	D₄	—	2	2	no	—
123	D₄	—	4	1	no	—
124	D₄	—	4	1	no	—
125	D₄	—	4	1	no	—
126	C₂×C₄	—	284	1	no	—
127	C₂×C₄	—	284	1	no	—
128	C₂×C₄	—	568	2	no	—
129	C₂×C₄	—	71	1	no	—
130	C₂×C₄	—	71	1	no	—
131	C₂×C₄	D₄×D₇₁	1	1	yes	characteristic
132	C₈	—	142	1	no	—
133	C₈	C₂×C₂²⋊D₇₁	1	1	yes	characteristic
134	C₈	#11765864	1	1	yes	characteristic
135	C₂²	—	4	2	no	—
136	C₂²	—	8	1	no	—
137	C₂²	—	8	1	no	—
138	C₂²	—	8	1	no	—
139	C₂²	—	8	1	no	—
140	C₂²	—	8	1	no	—
141	C₂²	D₈×D₇₁	1	1	yes	characteristic
142	C₄	—	568	1	no	—
143	C₄	—	568	1	no	—
144	C₄	—	142	1	no	—
145	C₄	—	142	1	no	—
146	C₄	#14556100	1	1	yes	characteristic
147	C₄	#14556152	1	1	yes	characteristic
148	C₂	—	8	2	no	—
149	C₂	—	16	1	no	—
150	C₂	#17986192	1	1	yes	characteristic
151	C₂	#17986428	1	1	yes	characteristic
152	C₂	#17986433	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#14556072

C₂

yes

maximal, characteristic, complemented

#17986382

C₂

yes

maximal, characteristic, complemented

D₈×Dic₇₁

C₂

yes

maximal, characteristic, complemented

Dic₇₁⋊C₁₆

C₂

yes

maximal, characteristic, complemented

#17985804

C₂

yes

maximal, characteristic, complemented

#17985800

C₂

yes

maximal, characteristic, complemented

C₂×D₁₆×C₇₁

C₂

yes

maximal, characteristic

C₂×Dic₂₈₄

C₂²

yes

characteristic

C₂×#11765853

—

#14556115

—

D₄×Dic₇₁

—

#14556072

—

C₂×C₇₁⋊C₁₆

C₂²

yes

characteristic

#14556062

C₂²

yes

characteristic

#14556049

—

C₈×Dic₇₁

C₂²

yes

characteristic

C₂×D₈×C₇₁

C₂²

yes

characteristic

C₂×D₈×C₇₁

C₂²

yes

characteristic

D₁₆×C₇₁

—

C₂×C₁₆×C₇₁

C₂²

yes

characteristic

C₂×Dic₁₄₂

—

C₂²×Dic₇₁

—

Dic₂₈₄

—

Dic₂₈₄

—

#11765853

—

C₂×C₇₁⋊C₈

D₄

yes

characteristic

C₄⋊Dic₇₁

—

C₂²⋊Dic₇₁

—

C₂²⋊Dic₇₁

—

C₂²⋊Dic₇₁

—

C₄×Dic₇₁

D₄

yes

characteristic

C₇₁⋊C₁₆

—

C₂×D₄×C₇₁

—

C₂×D₄×C₇₁

—

D₈×C₇₁

—

D₈×C₇₁

—

D₈×C₇₁

—

D₈×C₇₁

D₄

yes

—

C₂×C₈×C₇₁

C₂³

yes

characteristic

C₁₆×C₇₁

—

C₂³×C₇₁

—

C₂³×C₇₁

—

D₄×C₇₁

—

D₄×C₇₁

—

D₄×C₇₁

—

D₄×C₇₁

—

C₂×C₄×C₇₁

C₂×D₄

yes

characteristic

C₂×Dic₇₁

—

C₂×Dic₇₁

—

C₂×Dic₇₁

—

C₂×Dic₇₁

D₈

yes

characteristic

C₂×Dic₇₁

D₈

yes

characteristic

Dic₁₄₂

—

Dic₁₄₂

—

C₈×C₇₁

C₂×D₄

yes

characteristic

C₈×C₇₁

#41

yes

characteristic

C₇₁⋊C₈

—

C₂²×C₇₁

—

C₂²×C₇₁

—

C₂²×C₇₁

—

C₂²×C₇₁

—

C₂²×C₇₁

—

C₂²×C₇₁

—

C₂²×C₇₁

C₂×D₈

yes

C₄×C₇₁

yes

C₄×C₇₁

yes

Dic₇₁

—

Dic₇₁

—

Dic₇₁

—

Dic₇₁

—

C₂×C₇₁

—

C₂×C₇₁

—

C₂×C₇₁

yes

C₂×C₇₁

yes

C₂×C₇₁

yes

—

C₇₁

yes

characteristic, complemented, Sylow

C₂×SD₃₂

—

C₂×D₁₆

D₇₁

yes

characteristic

#495

—

C₄×D₈

—

C₄⋊C₁₆

—

#358

—

#357

—

C₂×Q₁₆

—

C₂×SD₁₆

—

142

—

C₂×D₈

C₂×D₇₁

yes

characteristic

C₂×D₈

C₂×D₇₁

yes

characteristic

#124

—

142

—

C₄×D₄

—

142

—

SD₃₂

—

142

—

D₁₆

—

C₂×C₁₆

—

C₂×C₁₆

C₂×D₇₁

yes

characteristic

#107

—

#102

—

142

—

C₄×C₈

—

C₂×Q₈

—

142

—

C₂×D₄

—

100

C₂×D₄

—

101

C₂²×C₄

—

284

—

102

Q₁₆

—

142

—

103

Q₁₆

—

142

—

104

SD₁₆

—

284

—

105

D₈

—

106

D₈

—

107

D₈

—

108

D₈

C₂²⋊D₇₁

yes

—

109

C₂×C₈

—

110

C₂×C₈

C₂²×D₇₁

yes

characteristic

111

C₄⋊C₄

—

142

—

112

C₂²⋊C₄

—

284

—

113

C₂²⋊C₄

—

284

—

114

C₂²⋊C₄

—

284

—

115

C₄²

—

116

C₁₆

—

117

C₁₆

—

142

—

118

C₂³

—

119

C₂³

—

120

Q₈

—

284

—

121

Q₈

—

284

—

122

D₄

—

123

D₄

—

124

D₄

—

125

D₄

—

126

C₂×C₄

—

284

—

127

C₂×C₄

—

284

—

128

C₂×C₄

—

568

—

129

C₂×C₄

—

130

C₂×C₄

—

131

C₂×C₄

D₄×D₇₁

yes

characteristic

132

C₈

—

142

—

133

C₈

C₂×C₂²⋊D₇₁

yes

characteristic

134

C₈

#11765864

yes

characteristic

135

C₂²

—

136

C₂²

—

137

C₂²

—

138

C₂²

—

139

C₂²

—

140

C₂²

—

141

C₂²

D₈×D₇₁

yes

characteristic

142

C₄

—

568

—

143

C₄

—

568

—

144

C₄

—

142

—

145

C₄

—

142

—

146

C₄

#14556100

yes

characteristic

147

C₄

#14556152

yes

characteristic

148

C₂

—

149

C₂

—

150

C₂

#17986192

yes

characteristic

151

C₂

#17986428

yes

characteristic

152

C₂

#17986433

yes

characteristic

	subgroup	G/H
center	C₂²	D₈×D₇₁
commutator	C₂×C₈×C₇₁	C₂³
Frattini	C₂×C₈	C₂²×D₇₁
Fitting	C₂×D₁₆×C₇₁	C₂
socle	C₂²×C₇₁	C₂×D₈
hypercenter	C₂×D₁₆	D₇₁
hypocenter	C₇₁	#2210
LCS	G ▷ C₂×C₈×C₇₁ ▷ C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×D₁₆

subgroup

G/H

center

C₂²

D₈×D₇₁

commutator

C₂×C₈×C₇₁

C₂³

Frattini

C₂×C₈

C₂²×D₇₁

Fitting

C₂×D₁₆×C₇₁

C₂

socle

C₂²×C₇₁

C₂×D₈

hypercenter

C₂×D₁₆

D₇₁

hypocenter

C₇₁

#2210

LCS

G ▷ C₂×C₈×C₇₁ ▷ C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁

UCS

C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×D₁₆

N	H ≅ G/N	type
1. C₂×#14556072	C₂	semidirect
2. #17986382	C₂	semidirect
3. D₈×Dic₇₁	C₂	semidirect
4. Dic₇₁⋊C₁₆	C₂	semidirect
5. #17985804	C₂	semidirect
6. #17985800	C₂	semidirect
77. C₇₁	#2210	semidirect

H ≅ G/N

type

1. C₂×#14556072

C₂

semidirect

2. #17986382

C₂

semidirect

3. D₈×Dic₇₁

C₂

semidirect

4. Dic₇₁⋊C₁₆

C₂

semidirect

5. #17985804

C₂

semidirect

6. #17985800

C₂

semidirect

77. C₇₁

#2210

semidirect

Subgroups — #22590393

Conjugacy classes of subgroups

Complements